Abc conjecture

Statement

For every $\epsilon >0$, there exists a constant $C_{\epsilon }$ such that for any three relatively prime integers $a,b,c$ such that:

$a+b=c$,

we have the inequality:

$\max \left(|a|,|b|,|c|\right)\leq C_{\epsilon }\prod _{p|abc}p^{1+\epsilon }$

where the indicated product is only over prime divisors of the product $abc$.

Related facts

Analogous facts over other rings

• Mason-Stothers theorem states that the analogous statement holds over polynomial rings over fields with absolute value replaced by degree -- in fact, we do not even need the $\epsilon$.

Weaker facts and conjectures

• Logarithmic lower bound on number of non-Wieferich primes
• The abc conjecture implies Fermat's last theorem for sufficiently large primes. Fermat's last theorem was proved by Wiles in 1994, though the abc conjecture is still open.